Entire solutions to the Swift--Hohenberg equation via variational approach

2024/04/07 by Sergey Kolonitskii, Kolonitskii, S. B., L. M. Lerman +3
Computer Science · #35B06 #35B45 #35J35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2404.05066

openalex publication_date 2024/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the stationary Swift--Hohenberg equation (Δ+ 1)² u - αu - βu² + u³=0 in the whole space \mathbb Rⁿ, 2≤ n ≤ 7. We develop and modify the variational approach introduced by Lerman, Naryshkin and Nazarov (2020) and obtain a series of periodic solutions with certain additional symmetries.

Related

• Existence and asymptotic behavior of nontrivial solutions to the Swift-Hohenberg equation
• Abundance of entire solutions to nonlinear elliptic equations by the variational method
• Nonexistence of local conservation laws for the generalized Swift-Hohenberg equation
• Localised states in an extended Swift-Hohenberg equation
• Continuation and bifurcations of periodic orbits and symbolic dynamics in the Swift-Hohenberg equation